Multigraphs with High Chromatic Index

Transcription

1 Multigraphs with High Chromatic Index by Jessica M. McDonald A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Combinatorics and Optimization Waterloo, Ontario, Canada, 009 c Jessica McDonald 009

2 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii

3 Abstract In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg s Theorem says that χ g o+1 (where χ denotes chromatic index, denotes maximum degree, and g o denotes odd girth). We characterize this bound by proving that for a connected multigraph G, χ (G) = g o+1 if and only if G = µc g o and (g o + 1) (µ 1) (where µ denotes maximum edge-multiplicity). Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that χ max{ ρ, + 1} (where ρ = max{ E[S] S 1 : S V, S 3 and odd}). For example, we refine Goldberg s classical Theorem by proving that χ max{ ρ, g o+3 }. Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with χ > g o+3. The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture. iii

4 Acknowledgements I would first and foremost like to thank Penny Haxell, my doctoral supervisor, for her expert guidance and support throughout my entire time at Waterloo. I would also like to thank the other members of my committee Bruce Richter, Nick Wormald, Therese Biedl, and Hal Kierstead for their helpful suggestions regarding the final draft of this thesis. Without making a huge list of names, I really would like to thank everyone in my life, here and away, who supported me throughout my graduate studies. Thanks especially to all the great friends that I met in Waterloo, for helping make my time here so enjoyable, and to my family, who are always there for me. iv

5 Contents List of Figures vii 1 Introduction 1 Edge-colourings, alternating paths and Tashkinov trees 5.1 Central results in edge-colouring From alternating paths to Tashkinov trees The proof of Tashkinov s Theorem Filling the gap As an algorithm Achieving maximum chromatic index Canonical examples Shannon s bound and Goldberg s bound Vizing s bound Bounding chromatic index A general result Specific new results Corresponding colouring algorithms Characterizing high chromatic index General characterization techniques High chromatic index with respect to g o Multiples of simple graphs and Vizing s bound v

6 6 Vertex-colouring Vertex-colouring results as edge-colouring results Edge-colouring results as vertex-colouring results Vertex-Tashkinov trees? Conclusion and future work 96 References 105 vi

7 List of Figures.1 The last vertex in a maximal φ-kierstead path P The case k(t )= Claim Working to establish (A) Working to establish (B) Working to establish (C) The location of the colours ψ(w) and ψ(v) Edges out of the triangle are limited in colour The path P The graph H The paths P 0 and P The forbidden subgraphs of Beineke s Theorem The forbidden subgraphs of Theorem Two multigraphs with the same line graph No maximal Tashkinov tree goes through the dotted line Every maximal Tashkinov tree is spanning A colouring modification using three alternating paths vii

8 Chapter 1 Introduction An edge-colouring of a multigraph G is an assignment of colours to the edges of G such that adjacent edges receive different colours. The central problem in this area is to determine the minimum number of colours needed for an edge colouring - this is called the chromatic index of the multigraph, and denoted χ := χ (G). It is easy to see that the chromatic index of a multigraph G must always be at least its maximum degree := (G). But how high can chromatic index be? If G is a simple graph, then the answer to this question is very easy. Due to a fundamental theorem of Vizing [44] from the 1960 s, the chromatic index of G must be equal to either or + 1. A great amount of research has gone into trying to distinguish these two chromatic classes, however in general the problem is known to be NP-hard [16]. In multigraphs, on the other hand, there is no such dichotomy. For example, the multigraph K 3 has = 4 but its chromatic index is clearly 6. In general, a multigraph may have chromatic index which greatly exceeds + 1. Such multigraphs are the subject of this thesis. There are limits to the value of chromatic index for multigraphs. Famous such upper bounds by Vizing [44] and Goldberg [11] will be of central importance in this thesis. These bounds combine and other graph parameters, namely maximum edge-multiplicity µ := µ(g) and odd-girth g o := g o (G) (the length of the shortest odd cycle in G), respectively. We will introduce and contextualize these two theorems in Chapter, where we also present the rest of the background information necessary for our work here. Paramount among this is a discussion of the celebrated Seymour- Goldberg Conjecture, and an introduction to the method of Tashkinov trees. The influence of the Seymour-Goldberg Conjecture on this thesis cannot be overstated. The conjecture, posed independently by Seymour [37] and by Goldberg [13] 1

9 Chapter 1: Introduction in the 1970 s, asserts that any multigraph G must have where χ (G) max { ρ(g), + 1} { } E[S] ρ(g) := max : S V (G), S 3 and odd. S 1 In our study we will support this conjecture in a number of ways, proving that multigraphs with highest chromatic index do obey this conjecture, and have chromatic index determined by a dense odd subgraph. In order to properly appreciate these results, it is essential that the reader keep the Seymour-Goldberg Conjecture in mind, beginning right from Chapter. The method of Tashkinov trees is the primary method we use to prove results in this thesis. This is a structural technique, developed by Tashkinov [4] in 000, which generalizes the earlier notion of Kierstead paths [1], and in turn the classical alternating path argument of König [4]. As such, Tashkinov trees lie at the heart of most major results in edge-colouring theory. There are however, some gaps in the English literature concerning this method. Hence, in Chapter, in addition to introducing Tashkinov trees, we provide both a proof and an algorithm to solidify the theory. In Chapter 3 we begin our study of multigraphs with high chromatic index by asking a natural question: given an upper bound on chromatic index, what type of multigraphs actually achieve this bound? That is, if we know that χ (G) a for all multigraphs G, can we characterize the chromatic class χ (G) = a? This chapter, which forms the basis of the paper [8], addresses this question for Goldberg s bound and for Vizing s bound. In the case of Goldberg s Theorem, we are able to get the complete characterization we seek (Theorem 3..). In the case of Vizing s Theorem however, we must settle for proving some necessary conditions of the chromatic class (Theorems ). In addition to studying known upper bounds, in this thesis we also want to provide new upper bounds for chromatic index in multigraphs. In Chapter 4 we show how Tashkinov trees can, in general, be used to prove upper bounds of the form χ max{ ρ, + t} for various values of t (Theorem 4.1.3). That is, we show how Tashkinov trees can be used to prove bounds which approximate the Seymour-Goldberg Conjecture. Of the particular bounds that we show using this method, Theorems 4..1 and 4.., involving odd-girth g o and girth g (the length of the shortest cycle in the underlying graph of G), respectively, actually prove that the Conjecture holds for a number

10 Chapter 1: Introduction of new classes of multigraphs (Theorems 4..3 and 4..4). Significantly, the bound involving g o also refines the classical upper bound of Goldberg. Our proofs in this chapter are algorithmic in nature, and we take care to provide an accompanying polynomial-time colouring algorithm for each of our specific bounds (Theorems ). Another way to interpret a result of the form is to think of it as χ max{ ρ, + t}, χ > + t χ = ρ. This is equivalent because ρ is actually a lower bound for chromatic index. To see this, note that for a multigraph G, given any odd set S V (G) with S 3, the maximum size of a colour class in G[S] is ( S 1)/. Hence, since chromatic index is an integer, E[S] χ (G[S]). S 1 We will use the fact that ρ is a lower bound for chromatic index repeatedly in this thesis in particular, we will use it whenever we want to establish the chromatic index of a particular multigraph. When we view our edge-colouring bounds in the form χ > + t χ = ρ, it is natural to ask if we can characterize the chromatic class χ > + t. We discuss this problem in Chapter 5 of this thesis. We first provide some general techniques for characterization, including a generalization of methods used in Chapter 3 (Propositions ). Then, we give a complete characterization for our g o result from the previous chapter (Theorem 5..1). As the Chapter 4 result refined Goldberg s bound, this characterization result extends the previous characterization of Goldberg s bound that we found in Chapter 3. Here in Chapter 5, we are also able to apply our general characterization techniques to a different result from Chapter 4, and get more information about Vizing s upper bound. More precisely, we are able to characterize large multiples of simple graphs that achieve Vizing s bound a result which is best-possible (Theorem 5.3.1). Edge-colouring a multigraph is equivalent to vertex-colouring its line graph, and in Chapter 6 we recognize this equivalence. We try to understand vertex-colouring results as they relate to edge-colouring, and vice versa. As all of our work in this thesis is based on the method of Tashkinov trees, we try to extend this method so 3

11 Chapter 1: Introduction that it can be used to vertex-colour graphs that are not line graphs of multigraphs. However, our efforts seem to indicate that such an extension is not possible. To conclude this thesis, we reflect on both the results that we have presented, and the methods we have used. This includes our contributions to understanding multigraphs with high chromatic index with respect to, µ, g, and especially g o. In this last chapter, we also look at the big picture and ask about the importance of Tashkinov trees in proving the Seymour-Goldberg Conjecture. We discuss other conjectures and problems in this area, how they are related to our work here, and future possibilities for study. As we shall see, the results of this thesis make significant headway in understanding multigraphs with high chromatic index - however they also raise a number of interesting questions. 4

12 Chapter Edge-colourings, alternating paths and Tashkinov trees In this chapter we introduce the background material necessary for the rest of this thesis. First, in Section.1, we describe the major known results in the area of edgecolouring. In Section., we define Tashkinov trees and state Tashkinov s Theorem. We show how many of the results in edge-colouring have been, or can be, attained using some form of Tashkinov tree. The third and final section of this chapter is devoted to establishing the proof of Tashkinov s Theorem and the algorithm implied by the proof. This is important, as we are aware of only one proof in English and it contains a flaw (not found in Tashkinov s original Russian publication). Also, while the algorithm implied by the proof has been alluded to by other authors, it has never been described explicitly, and we will need to build on this algorithm to get our colouring algorithms of Section Central results in edge-colouring Edge-colouring first appeared in graph theory literature in the 1880 s, courtesy of P. G. Tait ([40],[41], see also [10]). Tait was attempting to prove the Four Colour Theorem, and had shown an equivalence between 4-colourablity of planar maps, and 3-edge-colourability of cubic planar maps. Unfortunately, Tait s attempt to prove that every planar cubic map is 3-edge-colourable was seriously flawed. Given this rocky start, we might say that edge-colouring theory really began with the first correct result published König s Theorem of Theorem.1.1. [4] (König s Theorem) Let G be a bipartite multigraph. Then, χ (G) =. 5

13 Chapter : Edge-colourings,... König s Theorem is significant because it exhibits a large family of multigraphs which achieve the canonical lower bound for chromatic index. It took another thirty years for the first meaningful upper bound on chromatic index to be published. Now known as Shannon s Theorem, this result emerged through C. E. Shannon s [38] study of electrical networks. Theorem.1.. [38] (Shannon s Theorem) Let G be a multigraph. Then, χ (G) 3. A great breakthrough in edge-colouring theory arguably the great breakthrough when it comes to simple graphs was made by V. G. Vizing [44] in the 1960 s, with the following now-famous theorem. Theorem.1.3. [44] (Vizing s Theorem) Let G be a multigraph. Then, χ (G) + µ. If G is a simple graph, then µ = 1 and Vizing s Theorem gives just the two possibilities for chromatic index and + 1. A great amount of the work that has been done on edge-colouring has been to try to distinguish graphs of class one (chromatic index ) and class two (chromatic index + 1). However, in 1981, Holyer [16] proved that this is an NP-hard decision problem. Keeping this in mind, in this thesis we will not make any attempt to distinguish between these two lowest possible values for chromatic index in multigraphs our interest lies in multigraphs with high chromatic index. The innovation of Vizing s Theorem led to increased research in edge-colouring. Two important results which have emerged since are a refinement of Shannon s bound, due to Goldberg [11], and a refinement of Vizing s bound, due to Steffen [39]. These results use the concepts of odd-girth g o and girth g, which we should note are only defined in multigraphs which contain an odd cycle, or contain a cycle, respectively. Restricting ourselves to such multigraphs is not a major assumption however, as we already know that all bipartite multigraphs have chromatic index exactly. Theorem.1.4. [11] (Goldberg s Theorem) Let G be a multigraph containing an odd cycle. Then, χ (G) g o 1. Theorem.1.5. [39] (Steffen s Theorem) Let G be a multigraph containing a cycle. Then, µ χ (G) +. g/ 6

14 Chapter : Edge-colourings,... Since girth and odd-girth are both at least three, it is easy to see how Goldberg s bound refines Shannon s bound, and how Steffen s bound refines Vizing s bound. Another way of refining Vizing s bound is the following early theorem of Ore [30]. Here, rather than using the global parameters of maximum degree and maximum edge-multiplicity, we have the local parameters d(v), the degree of a vertex v, and µ(v), the maximum multiplicity of an edge incident to v. Theorem.1.6. [30] (Ore s Theorem) Let G be a multigraph. Then, χ (G) max{d(v) + µ(v) v V (G)}. In addition to the above upper bounds, the last forty years has also seen a number of conjectures emerge, each purporting to explain edge-colouring in more depth. The most significant of these was proposed independently by Seymour [37] and by Goldberg [13] in the 1970 s, and is now referred to as the Seymour-Goldberg Conjecture. This conjecture, already discussed briefly in our introduction, can be stated in the following three forms: and χ max{ ρ, + 1}, χ > + 1 χ = ρ, χ { ρ,, + 1}. This last version highlights the great strength of the Seymour-Goldberg Conjecture: it generalizes Vizing s Theorem for simple graphs by showing that there are exactly three possible values for the chromatic index of a multigraph. Moreover, while we know that it is NP-hard to decide between chromatic index and + 1, if the Seymour-Goldberg Conjecture is true, then deciding whether or not χ (G) > + 1 (and determining chromatic index exactly in this case) is polynomial-time solvable. This is because Edmonds Matching Polytope Theorem [9] implies that max{ρ(g), } is equal to the fractional chromatic index of G, a quantity that can be computed in polynomial time. (See, for example, [] for more on fractional chromatic index). Not only is the Seymour-Goldberg Conjecture the most important conjecture in edge-colouring, but it is strong enough to imply many other open conjectures in the area. This list includes both the Critical Multigraph Conjecture the Weak Critical Graph Conjecture, which we will discuss later in this thesis. One fact which has been firmly established however, is that the Seymour-Goldberg Conjecture is true asymptotically. That is, Kahn [0] has shown that for a multigraph G, χ (G) max{ρ(g), } as max{, ρ(g)}. 7

15 Chapter : Edge-colourings,... The conjecture is also known to be true for all multigraphs that do not contain a K5 -minor (Marcotte [7]). In addition, there is family of approximation results, starting in the 1970 s, which prove that { χ max ρ, } m 1 for certain values of m. Such results have been proved by Goldberg [11][1] (m = 9), Nishizeki and Kashiwagi [9] and independently Tashkinov [4] (m = 11), Stiebitz, Favrholt and Toft [8] (m = 13), and Scheide [35] (m = 15). Since 14 < 1 15, the best of these results show that the Seymour-Goldberg Conjecture holds when maximum degree is at most 15. One may note that the name Tashkinov appears in the above paragraph, and this is not a coincidence. The 000 citation given is indeed where Tashkinov trees were first introduced. However, as we shall see in the next section, the majority of results mentioned here used the method of Tashkinov trees in their proofs even going back to König s Theorem whether the authors were aware of it or not.. From alternating paths to Tashkinov trees The proof of König s Theorem is based on the idea of building a colouring one edge at a time, using alternating paths. Suppose we have a partial edge-coloring of a multigraph (with at least colours), but there is an edge e = xy still uncoloured. There must be at least one colour α not used on any edge incident to x, and at least one colour β not used on any edge incident to y. If α = β, then we can clearly colour e, and otherwise we consider the maximal (α, β)-alternating path beginning at x. We can swap the two colours along this path if we like, without affecting the fact that the colouring is proper. As long as the path does not end at y, such a swap would allow us to extend the colouring to e with the colour β. In the case of a bipartite graph, the path certainly cannot end at y, because this would create an odd cycle. This proves that colours are sufficient to edge-colour a bipartite graph, that is, this proves König s Theorem. While the idea of using alternating paths to augment partial edge-colourings seems very basic, it is really at the heart of many results in the edge-colouring literature. To better understand this, we need the much more general method of Tashkinov trees. 8

16 Chapter : Edge-colourings,... Let G be a multigraph and let φ be a partial edge colouring of G. We say that T = (p 0, e 0, p 1,..., p n 1, e n 1, p n ) is a φ-tashkinov tree in G if (T1.) p 0,..., p n are distinct vertices in G and, for each i {0,..., n 1}, e i E(G) and has ends p i+1 and p k for some k {0,..., i}, and (T.) e 0 is uncoloured by φ and for each i {1,..., n 1}, φ(e i ) j i φ(p j ), where φ(p j ) is the set of colours not appearing on any edge incident to p j. We will also refer to φ(p j ) as the set of colours that are not used at p i, that are not seen at p i, or, most commonly, that are missing at p i. Note that (T1) merely requires T to be a tree the weight of the definition is in (T), where we restrict the colour of each edge in the tree. We say that a φ-tashkinov tree T = (p 0, e 0, p 1,..., p n ) is φ-elementary if all the colours missing at its vertices are distinct, i.e., φ(p i ) φ(p j ) = for all 0 i < j n. It will also sometimes be convenient to refer to a set of vertices W as φ-elementary this means φ(w 1 ) φ(w ) = for all distinct w 1, w W. Clearly, an alternating path is always a Tashkinov tree. Moreover, an alternating path used to augment a colouring φ must have two vertices with a common missing colour (the two ends of the path), and so it is not φ-elementary. The following theorem describes this augmenting in the more general setting that is, it provides the framework for the method of Tashkinov trees. Note that given a partial edge-colouring φ, dom(φ) denotes the domain of φ, that is, the set of edges that are coloured by φ. Theorem..1. [4] (Tashkinov s Theorem) Let G be a multigraph and let φ be a partial ( + s)-edge-colouring of G, with s 1. Suppose that there exists a φ- Tashkinov tree T = (p 0, e 0, p 1,..., p n ) in G which is not φ-elementary. Then, there exists a ( + s)-edge colouring ψ of dom(φ) {e 0 }. We can now see how the method of Tashkinov s Theorem generalizes the alternating path technique of König. In either case we have a partial colouring, and a structure (alternating path or Tashkinov tree) with one uncoloured edge. We are able to modify the existing colouring so that it may be extended to the uncoloured edge, provided the structure has two vertices with a common missing colour. 9

17 Chapter : Edge-colourings,... Despite the similarities between alternating paths and Tashkinov trees, it should be emphasized that the jump from one to the other is quite large. In fact, there is no question that this jump would not have happened without one important intermediary step: Kierstead paths. The original proof of Vizing s Theorem involves building a fan structure where the first edge is uncoloured, and then uses a series of alternating paths to recolour and extend to the first edge. The fan structure here is not a Tashkinov tree however, and the recolouring process is somewhat complicated. In the 1980 s, H. A. Kierstead gave a new proof of Vizing s Theorem. Within his proof, Kierstead defined what we now refer to as Kierstead paths, which have the same definition as Tashkinov trees, except that in (T1), k is required to be i, making the structure a path. Kierstead proved Theorem..1 in the case that T is a path that is, he proved the following theorem. Theorem... [1] (Kierstead s Theorem) Let G be a multigraph and let φ be a partial ( + s)-edge-colouring of G, with s 1. Suppose that there exists a φ- Kierstead path P = (p 0, e 0, p 1,..., p n ) in G which is not φ-elementary. Then, there exists a ( + s)-edge colouring ψ of dom(φ) {e 0 }. Theorem.. is really the most difficult part of Kierstead s proof of Vizing s Theorem. Assuming this result, we can start the proof by assigning a maximum domain (χ 1)-edge colouring φ to G. If χ + 1 then Vizing s Theorem certainly holds. So, we may assume that (χ 1) + 1. Hence, we may apply Kierstead s Theorem. Choose any uncoloured edge e 0 with ends p 0 and p 1. Note that there must be some colour α missing at p 0, and that there must be an α-edge incident to p 1, which we may label as e 1. We continue to build this structure as far as we can, ending up with a maximal φ-kierstead path P = (p 0, e 0, p 1,..., p n ) in G which has n. Kierstead s Theorem tells us that P must be φ-elementary, since φ has maximum domain. So in particular, the last vertex in the path, p n, must see every colour missing at every previous vertex in the path. Since there are χ 1 colours being used, and each vertex may see coloured edges (except for p 0 and p 1, which see at most 1 each), we get that the number of colours seen by p n is at least n 1 φ(p i) i=0 n(χ 1 ) +. On the other hand, since P is maximal, each edge incident to p n that is coloured with one of these colours must be between p n and p 0,..., p n 1, as otherwise it could be used to extend P (See Figure.1). There are at most nµ such edges, so we must 10

18 Chapter : Edge-colourings,... n-1 U φ(pi ) i=0 p 0 p 1 p P p n-1 p n Figure.1: The last vertex in a maximal φ-kierstead path P have n(χ 1 ) + nµ χ 1 µ n χ + µ, proving Vizing s Theorem. Shannon s Theorem is even easier to obtain - just note that p n has degree at most, and get n(χ 1 ) + (χ 1 ) + χ 3. Of the other upper bounds in the previous section, Goldberg s bound was proved using the alternating paths technique that is generalized by both Kierstead paths and Tashkinov trees. Steffen used Kierstead paths explicitly in establishing his bound. Ore s bound can be easily proved by using a Tashkinov tree that is a fan (as noted by Favrholt, Stiebitz and Toft [8]). Of the results in the previous section towards the Seymour-Goldberg Conjecture, Kahn used the probabilistic method and Marcotte used a structural analysis but the others also used a Tashkinov tree method, in some form or another. We know that Tashkinov trees generalize Kierstead paths completely; however it is sometimes helpful to use Kierstead paths in particular, rather than the more general version. There are really two reasons for this. The first is that while a Kierstead path is always a Tashkinov tree, a maximal Kierstead path is not necessarily 11

19 Chapter : Edge-colourings,... a maximal Tashkinov tree. In proofs, it will always be important that the structure in question is as large as possible, in the sense that no other edge could be added. It may sometimes serve us to know that all (or part of) the Tashkinov tree we have constructed is actually an alternating path, a path, a fan, or some other specific structure. Note that in the case of the proofs of Vizing and Shannon s Theorems above, we did not need to appeal to such structure, and hence the arguments would have been identical had we constructed a maximal Tashkinov tree, instead of a maximal Kierstead path. However, there is another reason to use Kierstead paths instead of Tashkinov trees when possible they hide much less difficulty. We have already stated that the jump from alternating paths to Tashkinov trees is very large. As we shall see in the next section, however, the majority of this challenge lies in moving from Kierstead paths to Tashkinov trees..3 The proof of Tashkinov s Theorem We divide our work on the proof of Tashkinov s Theorem into two subsections. First, we concentrate on establishing the proof in English. Rather than presenting the entire (very lengthy) argument here, we refer the reader to [8], where a nearly complete proof is given, and simply endeavor to fill the gap. The (partial) proof that we present in our first subsection is based on a sequence of colour-swaps along alternating paths. In fact, the entire proof of Tashkinov s Theorem is like this, and naturally lends itself to an algorithm. While other authors have realized this (eg. [8], [35]), the algorithm has not yet been explicitly written out or analyzed - we provide these details in our second subsection..3.1 Filling the gap Tashkinov trees are generalizations of Kierstead paths and, as we have already mentioned, Tashkinov s Theorem is an extension of Kierstead s Theorem. Kierstead s original proof of his Theorem is easy to follow, and fits comfortably on a single page. The proof uses an induction on j and j i, where we begin by assuming that φ(p i ) φ(p j ) for some 0 i < j n. In Tashkinov s Theorem, we use an induction where we hope for trees to be as path-like as possible, so that we may eventually apply Kierstead s Theorem. Proof. (Theorem..1, Tashkinov s Theorem) Let G be a multigraph and let s 1, as in the statement of Theorem..1. However, suppose that there exists a counterexample to this Theorem. That is, suppose that there exists a partial ( +s)- edge colouring φ of G, and a φ-tashkinov tree T = (p 0, e 0, p 1,..., p n ) in G that is 1

20 Chapter : Edge-colourings,... T p 0 p 1 p = k(t) p n Figure.: The case k(t )= not φ-elementary, but such that dom(φ) {e 0 } is not ( + s)-edge colourable. Note that there exists k {0,..., n} such that T k := (p k,..., p n ) is a path; define k(t ) to be the smallest such k for T. Among all counterexamples, choose (T, φ) such that 1. k(t ) is as small as possible. V (T ) is as small as possible, subject to (1). Consider the value k(t ). Note that since e 0 = (p 0, p 1 ), it is impossible to have k(t ) = 1. If k(t ) = 0, then T is a φ-kierstead path. If k(t ) =, then (p 1, e 0, p 0, e 1, p,..., p n ) is a φ-kierstead path (See Figure.). In either case, since T is not φ-elementary, Kierstead s Theorem (Theorem..) says that dom(φ) {e 0 } is ( + s)-edge colourable, contradicting the fact that (T, φ) is a counterexample. So, we may assume that k(t ) 3. If k(t ) < n, then a correct argument to conclude this proof may be found in [8]. We will deal with the remaining case, that is, when k(t ) = n. We are searching for a contradiction to our counterexample (T, φ). Note that any truncation of T is still a φ-tashkinov tree. That is, if we define T j := (p 0, e 0, p 1,..., p j ), then we know that T j is a φ-tashkinov tree, for all j {1,..., n}. Moreover, k(t j ) k(t ) and V (T j ) < V (T ) for all j < n. So, it must be the case that every T j is φ-elementary, otherwise we would not have chosen (T, φ) as our counterexample. In particular, this implies that T n 1 is φ-elementary. Since T is not φ-elementary, this means that we must have φ(p i ) φ(p n ) for some i {0,..., n 1}. In addition to choosing T carefully for our counterexample, we also choose φ carefully. While T is a φ-tashkinov tree, it may also be a ψ-tashkinov tree, for 13

21 Chapter : Edge-colourings,... some other partial ( + s)-edge colouring ψ of G. Let C T denote the set of all colourings ψ such that (T, ψ) is a counterexample. Clearly, φ C T. Just as we know that T n 1 is φ-elementary, we also know that T n 1 is ψ-elementary, for every ψ C T. So, since (T, ψ) is a counterexample for all ψ C T, there must always be some i {0,..., n 1} such that ψ(p i ) ψ(p n ). We will proceed by proving the following series of statements: (A). There exists ψ C T such that ψ(p j ) ψ(p n ) for some j n 1, and some colour β ψ(p j ) ψ(p n ) is not used on any edge of T. (B). There exists ψ C T such that ψ(e n 1 ) is seen by p n 1. (C). There exists ψ C T such that ψ(e n 1 ) is seen by p n 1, and ψ(p j ) ψ(p n ) for some j n 1. Statement (A) will help us prove statement (B), and statement (B) will help us prove statement (C). Once we establish statement (C), we will be able to deduce a contradiction. The following two claims will aid us greatly in our arguments. We include both proofs as we will want to refer to them when we discuss the algorithm implied by this theorem. Claim 1. [8] Let ψ C T. If j {1,..., n 1}, then there are at least four colours in j i=0 ψ(p i) that are unused on the edges of T j. Proof of Claim. Since ψ(p 0 ), ψ(p 1 ) s + 1 and ψ(p i ) s for all i {,..., j}, and since T j is ψ-elementary, we know that j ψ(p i) i=0 (j + 1)s + j + 3. On the other hand, T j consists of exactly j edges, j 1 of which are coloured. Since (j + 3) (j 1) = 4, we get our desired result. Claim. [8] Let ψ C T. Suppose that α ψ(p i ), β ψ(p j ), and α is unused on the edges of T j, for some 0 i < j n 1. Then, α β, and there is a maximal (α, β)-alternating path Q between p i and p j (with respect to ψ). Moreover, if ψ = ψ(i, j, α, β) is the colouring obtained from ψ by switching α and β along Q, then ψ C T. 14

22 Chapter : Edge-colourings,... T j p 0 ψ α p i β α p j α β β ψ' = ψ(i, j, α, β) p 0 T j β p i α β p j β α α Figure.3: Claim Proof of Claim. It is clear that α β, because we know that T n 1 must be ψ- elementary. Let Q be the maximal (α, β)-alternating path starting at p j. Since T j is ψ-elementary, if Q ends at a vertex on T j, then it must end at p i. Moreover, the fact that T j is a ψ-elementary ψ-tashkinov tree implies that β cannot occur on an edge of T j. So, since α was chosen to be unused on the edges of T j, we have that E(Q) E(T j ) =. Because of this, if we define ψ from ψ by swapping α and β along Q, we know that T j is ψ -elementary. If Q does not end at p i, then α ψ (p i ) ψ (p j ), so T j is not ψ -elementary, which contradicts our choice of (T, ψ). So, Q must end at p i, and hence β ψ (p i ) and α ψ (p j ). Since T j has no α or β edges under ψ (or ψ), this is enough to tell us that T is a ψ -Tashkinov tree. Moreover, the fact that T is not ψ-elementary implies that T is not ψ -elementary, so ψ C T. Figure.3 depicts the colour change described in Claim, and it may be helpful to refer to this diagram when the claim is applied. Note that in Figure.3, and from here on in this thesis, a circle around a colour name means that the colour is missing at a particular vertex. Also, we often draw a wavy line for a tree, to indicate that while it may help to think of the tree as a path, it may in fact have a more complicated structure. 15

23 Chapter : Edge-colourings,... T p 0 α p m β p j p n β Figure.4: Working to establish (A) We now work to establish (A). To this end, let ψ be any colouring in C T. We know that there exists β ψ(p j ) ψ(p n ) for some j {0,..., n 1}, by our discussion above. Claim 1 tells us that there are at least 4 colours in n i=0 ψ(p i) that are unused on the edges of T n (since n = k(t ) 3). There are only two edges in T that are not in T n, so we may in fact choose a colour α n i=0 ψ(p i) that is unused on T, and that also satisfies α β. Say α ψ(p m ). We now have the situation depicted roughly in Figure.4, and will proceed according to the value of j. Suppose first that j = n 1. We claim that in this case, β is unused on T. To see this, first note that β ψ(p n ) implies that β is not used on the last edge of T. On the other hand, if β is used on an edge of T n 1, then it must be missing at some vertex in T n (by definition of a Tashkinov tree). Since we are assuming that β ψ(p n 1 ) already, this would mean that T n 1 is not ψ-elementary, a contradiction. So, indeed, β is not used on any edge of T. Since α is unused on T, we can modify ψ to get ψ = ψ(m, j, α, β) C T, as described in Claim. Then, β ψ (p m ) ψ (p n ). However, since neither α nor β were used on T under φ, we also get that β is not used on T under ψ. Hence, ψ satisfies (A) in this case. Suppose now that j n (refer again to Figure.4). If (A) is not satisfied, then it must mean that β is used on T. We may assume that α is seen by p n, since if not, the fact that α is also missing at p m and that α is unused on T immediately implies that ψ satisfies (A). So, we may consider the maximum (α, β)-alternating path Q starting at p n. If Q contains a vertex on T n, then let Q be the segment of Q between p n and the first vertex of Q on T n, and define T = (p 0, e 0, p 1,..., p n, Q, p n ). Then, T is a ψ-tashkinov tree (since α and β are missing at vertices in T n ), and T is not ψ-elementary, (since ψ(p j ) ψ(p n ) ). However, k(t ) n 1, so (T, ψ) contradicts our choice of (T, φ). Hence, we know that Q cannot contain any vertices from T n. This implies that Q does not contain any edges of T either, since both e n 1 and e n have one end in T n (for e n 1 this is because k(t ) = n). With this in mind, let ψ be the colouring obtained by switching the colours β and α along Q. 16

24 Chapter : Edge-colourings,... p 0 T p j β e n-1 γ γ p n-1 p n β Figure.5: Working to establish (B) p 0 T p m α e n-1 γ β p n-1 p n β Figure.6: Working to establish (C) After this switch, we get α ψ (p m ) ψ (p n ), and α is unused on T. So, ψ satisfies (A). We have now established (A), and want to work to prove (B). Let ψ C T be a colouring satisfying (A), i.e., with ψ(p j ) ψ(p n ) for some j n 1, and some colour β ψ(p j ) ψ(p n ) not used on any edge of T. If ψ does not satisfy (B), then it means that ψ(e n 1 ) ψ(p n 1 ). For ease of notation, let γ := ψ(e n 1 ). We have the situation depicted by Figure.5. Since β is unused on T, we may define ψ = ψ(j, n 1, β, γ) as in Claim, and be assured that ψ C T. However now, p n 1 sees γ under ψ, and we know that ψ (e n 1 ) = γ (since β ψ(p n )). So, we have succeeded in finding a colouring in C T satisfying (B). Suppose now that there exists ψ C T satisfying (B) but not (C). This means that while γ := ψ(e n 1 ) is seen by p n 1, we must have ψ(p i ) ψ(p n ) = for all i n 1. So, since T is not ψ-elementary, there must exist some β ψ(p n 1 ) ψ(p n ). Claim 1 tells us that there are at least 4 colours in n i=0 ψ(p i) which are unused on the edges of T n (since n = k(t ) 3). So, we may choose one of these colours α, say α ψ(p m ), that is also unused on T. Since γ is used on T, clearly α γ. We also know that α β, because otherwise T n 1 would not be ψ-elementary. We have the situation depicted in Figure.6. Now we may define a new colouring ψ = ψ(m, n 1, α, β), as described in Claim, and be assured that ψ C T. Since γ α, β, we know that ψ still satisfies (B). However, ψ also satisfies (C), since β ψ (p m ) ψ (p n ). 17

25 Chapter : Edge-colourings,... We have now succeeded in finding a colouring ψ C T satisfying (C), and we will use this ψ to get a contradiction. To this end, we define T = (p 0, e 0, p 1,..., p n, e n 1, p n ). We claim that T is a ψ-tashkinov tree. Certainly, since k(t ) = n, we know that e n 1 = (p i, p n ) for some i n. Also, T is a ψ-tashkinov tree, so since ψ(e n 1 ) is seen by p n 1 (by (C)), this colour must be missing at some p l for l n. Hence, T satisfies the two properties of being a ψ-tashkinov tree. However, by (c), T is not ψ-elementary, so, (T, ψ) is a counterexample. Since k(t ) < k(t ), this contradicts our choice of (T, φ), and hence completes our proof..3. As an algorithm We shall refer to the algorithm implied by Tashkinov s Theorem as Tashkinov s algorithm. Here, we will not only detail Tashkinov s algorithm, but we will prove that in general, it runs in polynomial time. In fact, we shall see that we can modify the algorithm slightly so that the number of recolourings required only depends on. Before we can even start to discuss the specifics of Tashkinov s algorithm however, we need to discuss Kierstead s algorithm. Although we have not seen all of the proof of Tashkinov s Theorem, we did see that Kierstead s Theorem is an essential part of the proof. So, it should not be a surprise that Tashkinov s algorithm is dependent on the algorithm implied by Kierstead s proof, which we call Kierstead s algorithm. This much simpler algorithm has the following input and output. Kierstead s Algorithm INPUT: multigraph G partial ( + s)-edge colouring φ of G (s 1) φ-kierstead path P = (p 0, e 0, p 1,..., p n ) that is not φ-elementary OUTPUT: ( + s)-edge colouring φ of G with dom(φ ) = dom(φ) {e 0 } 18

26 Chapter : Edge-colourings,... Kierstead s algorithm has never been explicitly detailed; however it is an important part of Tashkinov s algorithm, and hence an important part of our algorithms later in this thesis. So, we take the time to provide the details now. Our description follows Kierstead s proof exactly, and hence the reader may refer to [1] for additional clarifications as necessary. To begin Kierstead s algorithm, initialize i < j such that φ(p i ) φ(p j ). Then, the algorithm is iterative. In each iteration we modify φ and/or P so that we can choose new i, j with φ(p i ) φ(p j ), and j strictly smaller than it previously was, or j i strictly smaller than it previously was. When we get to i = 0 and j = 1, then we can define φ. At the start of an iteration, we have φ, a φ-kierstead path (p 0, e 0, p 1,..., p N ) (where N n), and values for i < j such that φ(p i ) φ(p j ). We begin the iteration by doing the following: 0. Choose α φ(p i ) φ(p j ). We then proceed according to which of the following three cases we are in: Case 1: j = 1 Case : j > 1 and j i = 1 Case 3: j > 1 and j i > 1 Case 1 is the easiest to resolve. Here, there is only one step we need to do: 1. Let φ be the (partial) edge-colouring obtained from φ by colouring e 0 with α. Once we have done this, we are not only finished with the iteration, but we have completed the algorithm. We stop and output φ. Case is only slightly more complicated. Here, we proceed as follows: 1. Find p m such that β := φ(e j 1 ) φ(p m ). (Such an m < j 1 exists by definition of Kierstead path).. Modify φ by recolouring edge e j 1 with α, and modify P by truncating after p j 1. (P is still clearly a φ-kierstead path). 3. Set i := m and j := j 1, and proceed to the next iteration, having decreased the value of j. (This is allowed because now, β φ(p m ) φ(p j 1 )). Case 3 is the most involved case for Kierstead s algorithm. We proceed as follows: 19

27 Chapter : Edge-colourings, Choose δ φ(p i+1 ). (This is possible because s 1).. If δ = α, set j := i + 1 (which decreases j i), and then proceed to the next iteration. 3. Consider C, the maximal (α, δ)-alternating path starting at p i+1. (a) If C ends at a vertex p k with k < i, then set i := k and j := i or i + 1 (depending on whether α or δ is missing at p k ). Proceed to the next iteration, having decreased the value of j. (b) If C contains an edge e k with k i, then find a vertex p l with l < k and φ(e k ) φ(p l ) (such an l exits by definition of Kierstead path). Set i := l, j := i or i + 1, (depending on whether φ(e k ) is α or δ). Proceed to the next iteration, having decreased the value of j. (c) Otherwise, modify φ by swapping α and δ along C. (Note that this makes α φ(p i+1 )). i. If C ends at p i, then set i := i + 1 and leave j unchanged. Proceed to the next iteration, having decreased the value of j i. ii. If C does not end at p i, then modify P by truncating after p i+1. Then, leave i unchanged, but set j := i + 1. Proceed to the next iteration, having decreased the value of j. The important fact to realize about our description of Kierstead s algorithm is that each iteration does terminate, and hence the algorithm will terminate. Moreover, note that each iteration consists of at most one re-colouring. The worst-case scenario for this algorithm is that we start with (i, j) = (0, n), and then it takes n iterations to get to (n 1, n), then one iteration to get to (n, n 1), then n 1 iterations to get to (n, n 1), then one iteration to get to (n 3, n ), and so on, until we get to (0, 1). In total, this would mean n(n + 1) iterations. Since n V (T ) 1, we clearly need only a polynomial number of recolourings (depending on V (G) ). In fact, we can ensure that the number of recolourings depends only on, if we add in one preliminary step. The preliminary step is to truncate P after p 1. The reason we can do this, and still have our algorithm work (that is, still be able to pick α in Step 0), is as follows. If P is any φ-elementary φ-kierstead path, for φ a partial colouring with + s colours, then + n 0

28 Chapter : Edge-colourings,... the union of colours missing on P has size at most + s. So, V (P ) ( + s ) + + s s V (P ) + + s s( V (P ) 1) V (P ) + 1. s Since s 1, this means that V (P ) 1. So, every φ-kierstead path with at least vertices is not φ-elementary. Hence, the preliminary step works as desired. Note that, although the number of recolourings depends only on, many of these recolourings involve swapping an alternating path, which could involve all the vertices of the multigraph. Hence, overall, the algorithm is polynomial depending on V (G) and. Now that we have seen Kierstead algorithm, we are ready to detail Tashkinov s algorithm. Like Kierstead s algorithm, Tashkinov s algorithm is iterative, however the induction parameters are quite different. Just as we saw in the proof of Tashkinov s Theorem, the goal now is to make our Tashkinov tree as path-like as possible, and then apply Kierstead s algorithm. In each iteration, we modify T and/or φ so that either k(t ) decreases (where k(t ) is defined as in the proof of Tashkinov s Theorem), or k(t ) remains the same and V (T ) decreases. The input and output of the algorithm are as follows. Tashkinov s Algorithm INPUT: multigraph G partial ( + s)-edge colouring φ of G (s 1) φ-tashkinov tree T = (p 0, e 0, p 1,..., p n ) that is not φ-elementary OUTPUT: ( + s)-edge colouring φ of G with dom(φ ) = dom(φ) {e 0 } Suppose that we start an iteration of Tashkinov s Algorithm with the tree T = (p 0,..., p N ). (Note that if this is not our first iteration, then N may not be equal to n). We first check our two induction parameters. 1

29 Chapter : Edge-colourings, (a) If k(t ), then either T is a φ-kierstead path or can be reordered as a φ-kierstead path (see Figure.). So, apply Kierstead s algorithm to get φ. This completes not only the iteration, but the entire algorithm, so stop and output φ. (b) If T N 1 is not φ-elementary, then replace T with T N 1, and proceed to the next iteration. (T N 1 := (p 0,..., p N 1 ), as defined in the proof of Tashkinov s Theorem). With this preliminary step taken care of, we proceed with the iteration depending on which of the following three cases T falls into. Case 1: k(t ) = N Case : k(t ) < N and i {k(t ),..., N 1} such that φ(p i ) φ(p N ) Case 3: k(t ) < N and φ(p i ) φ(p N ) = i {k(t ),..., N 1} Unlike Kierstead s algorithm, where only one case involved swapping colours along an alternating path, all three cases here are potentially complex. Since we have already seen the proof for Case 1, it should be the easiest to follow. However, there is an important issue to note regarding Claims 1 and from the proof of Tashkinov s Theorem. Looking at the short proof of Claim 1 that we presented, it is easy to see that it will hold at every point in our algorithm. However, Claim may not hold at all times during the algorithm. Certainly, if we have the set-up of the claim with T and φ, and if Q is the maximal (α, β)-alternating path starting at p j, then we can define φ = φ(i, j, α, β) by swapping α and β along Q. Moreover, because of our inclusion of Step 0(b), we may assume that T n 1 is φ-elementary, which is a key element of our proof. The only potential problem with the proof of Claim then, is that it is possible that T j is not φ -elementary. This was not possible in our proof of Claim, because there, T and φ had already been chosen as the best possible counterexample which is essentially what we are striving for in this algorithm. So, this problem actually represents another desirable outcome for us, where we could immediately proceed to the next iteration with T and φ replaced by T j and φ. So, in our algorithm, each time we want to replace φ with φ(i, j, α, β), we either get that T j is not elementary with respect to the new colouring (and we can immediately complete the iteration), or we get that Claim holds. Now we return to our description of an iteration of Tashkinov s algorithm. If we are in Case 1, then we proceed as follows: 1. Choose j and β such that β φ(p j ) φ(p N ).

30 Chapter : Edge-colourings,.... We want to ensure that φ and β satisfy condition (A) from the proof, namely j N 1, and β is not used on any edge of T. If this holds, then go to the next step. If not, choose m N and α φ(p m ) such that α is not used on the edges of T N (possible by Claim 1 ). If α φ(p N ), swap the names of α and β, set j := m, and proceed to the next step. Otherwise, do as follows: (a) If j = N 1, replace φ by φ(m, j, α, β). If T j is not φ-elementary with respect to this new colouring, then replace T with T j and proceed to the next iteration. Otherwise, set j := m and proceed to the next step, knowing that (A) is satisfied. (b) If j N but β is used on T, let Q be the maximal (α, β)-alternating path starting at p N. i. If Q contains a vertex on T N, then let Q be the segment of Q between p N and the first vertex of Q on T N, and define T = (p 0,..., p N, Q, p N ). Proceed to the next iteration with T in place of T. (Possible because k(t ) < N 1). ii. Otherwise, modify φ by switching the colours α and β along Q. Then switch the names of α and β throughout G, to get that φ satisfies (A). 3. We now want to ensure that φ satisfies condition (B) from the proof, namely γ := φ(e N 1 ) is seen by p N 1. If this holds, then go directly to the next step. If not, replace φ by φ(j, N 1, β, γ). If T N 1 is not φ-elementary, then replace T with T N 1 and proceed to the next iteration. Otherwise, φ satisfies (B). 4. We now want to ensure that φ satisfies condition (C) from the proof, namely γ = φ(e N 1 ) is seen by p N 1, and φ(p j ) φ(p N ) for some j N 1. (Note that this first condition already holds.) Pick β φ(p j ) φ(p N ) for some j (possible since T N 1 is φ-elementary). If j N 1 then proceed to the next step. If j = N 1, choose α φ(p m ) that is unused on T, for some m N (possible by Claim 1). Then, replace φ with φ(m, N 1, α, β ). If T m is not φ-elementary with respect to this new colouring, then replace T with T m and proceed to the next iteration. Otherwise, proceed to the next step, knowing that φ satisfies (C). 3